Label  

Description

Maj

The current major iteration number.

Mnr

is the number of iterations required by both the feasibility and optimality phases of the QP subproblem.  Generally,  Mnr will be 1 in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Description of the method).

Step

The step length a taken along the current search direction p. 
The variables x  have just been changed to x + ap.  On reasonably well-behaved problems, the unit step will be taken as the solution is approached.

nObj

The number of times subroutine funobj has been called to evaluate the nonlinear objective function. Evaluations needed  for the estimation of the gradients by finite differences are  not included.  nObj is printed as a guide to the amount of  work required for the linesearch.

nCon

The number of times subroutine funcon has been called to evaluate the nonlinear constraint functions.

Merit

is the value of the augmented Lagrangian merit function (see Major iterations). 
This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Description of the method).  As the solution is approached, Merit will converge to the value of the objective at the solution.

In elastic mode, the merit function is a composite function involving the constraint violations weighted by the elastic weight.

If the constraints are linear, this item is labeled Objective, the value of the objective function.   It will decrease monotonically to its optimal value.

Feasbl

is the value of rowerr, the maximum component of the scaled nonlinear constraint residual (Specs file checklist and defaults).  The solution is regarded as acceptably feasible if Feasbl is less than the Major feasibility tolerance.

 If the constraints are linear, all iterates are feasible and this entry is not printed.

Optimal

is the value of maxgap, the maximum complementarity gap (Subroutine snInit).  It is an estimate of the degree of nonoptimality of the reduced costs.  Both Feasbl and Optimal are small in the neighborhood of a solution.

nS

The current number of superbasic variables.

Penalty

is the Euclidean norm of the vector of penalty parameters used in the augmented Lagrangian merit function (not printed if nnCon is zero).

LU

The number of nonzeros representing the basis factors L  and U on completion of the QP subproblem.

If nonlinear constraints are present, the basis factorization B = LU  is computed at the start of the first minor iteration. At this stage, LU = lenL + lenU,  where lenL, the number of subdiagonal elements in the columns of a lower triangular matrix and lenU is the number of diagonal and superdiagonal elements in the rows of an upper-triangular matrix.

As columns of B are replaced during the minor iterations, LU may fluctuate up or down but in general will tend to increase.   As the solution is approached and the minor iterations decrease  towards zero, LU will reflect the number of nonzeros in the LU factors at the start of the QP subproblem.

If the constraints are linear, refactorization is subject only to the Factorize frequency, and LU will tend to increase between factorizations.

Swp

The number of columns of the basis matrix B  that were swapped with columns of S to improve the condition of B .  The swaps are determined by an LU factorization of the rectangular matrix B_S = (B  S)T with stability being favored  more than sparsity.

Cond Hz

An estimate of the condition number of RTR, an estimate of ZTHZ, the reduced Hessian of the Lagrangian.   It is the square of the ratio of the largest and smallest diagonals of the upper triangular matrix R  (which is a lower bound on the condition number of RTR.  Cond Hz gives a rough indication of whether or not the optimization procedure is having difficulty.  If e is the relative precision of the machine being used, the SQP algorithm will make slow progress if Cond Hz becomes as large as e-½ 108, and will probably fail to find a better solution if Cond Hz reaches e-3/4 1012.

To guard against high values of Cond Hz , attention should be given to the scaling of the variables and the constraints.  In some cases it may be necessary to add upper or lower bounds to certain variables to keep them a reasonable distance from singularities in the nonlinear functions or their derivatives.

PD

is a two-letter indication of the status of the convergence tests involving primal and dual feasibility of the iterates (see Specs file checklist and defaults and Subroutine snInit) in the description of Major feasibility tolerance and Major optimality tolerance.  Each letter is T if the test is satisfied, and F otherwise.

If either of the indicators is F when snopt terminates with inform = 0, the user should check the solution carefully.

 Code

Meaning

c

Central differences have been used to compute the unknown components of the objective and constraint gradients.  A switch to central differences is made if either the linesearch gives a small step, or x is close to being optimal.  In some cases, it may be necessary to re-solve the QP subproblem with the central-difference gradient and Jacobian.

d

During the linesearch it was necessary to decrease the step in order to obtain a maximum constraint violation conforming to the value of Violation limit.

l

The norm-wise change in the variables was limited by the value of the Major step limit.  If this output occurs repeatedly during later iterations, it may be worthwhile increasing the value of Major step limit.

i

If SNOPT is not in elastic mode, an "i'' signifies that the QP subproblem is infeasible.  This event triggers the start of nonlinear elastic mode, which remains in effect for all subsequent iterations.  Once in elastic mode, the QP subproblems are associated with the elastic problem NP(g).

 If SNOPT is already in elastic mode, an "i'' indicates that the minimizer of the elastic subproblem does not satisfy the linearized constraints.  (In this case, a feasible point for the usual QP subproblem may or may not exist.)

M

An extra evaluation of the problem functions was needed to define an acceptable positive-definite quasi-Newton update to the Lagrangian Hessian.  This modification is only done when there are nonlinear constraints.

m

This is the same as "M'' except that it was also necessary to modify the update to include an augmented Lagrangian term.

R

The approximate Hessian has been reset by discarding all but the diagonal elements.  This reset will be forced periodically by the Hessian frequency and Hessian updates keywords.
However, it may also be necessary to reset an ill-conditioned Hessian from time to time.

r

The approximate Hessian was reset after ten consecutive major iterations in which no BFGS update could be made.  The diagonals of the approximate Hessian are retained if at least one update has been done since the last reset.  Otherwise, the approximate Hessian is reset to the identity matrix.

s

A self-scaled BFGS update was performed. This update is always used when the Hessian approximation is diagonal, and hence always follows a Hessian reset.

S

This is the same as a "s'' except that it was necessary to modify the self-scaled update to maintain positive definiteness.

n

No positive-definite BFGS update could be found. The approximate Hessian is unchanged from the previous iteration.

f

the Hessian was reset to the identity matrix under control of the keyword Hessian flush. This can happen only once in the course of the minimization.

t

The minor iterations were terminated at the Minor iteration limit.

u

The QP subproblem was unbounded.

w

A weak solution of the QP subproblem was found.